design of rotating electrical machines - malestrom

DESIGN OF ROTATING ELECTRICAL MACHINES

Design of Rotating Electrical Machines Juha Pyrh onen, Tapani Jokinen and Valria Hrabovcov e a 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-69516-6

DESIGN OF ROTATING ELECTRICAL MACHINESJuha Pyrh nen oDepartment of Electrical Engineering, Lappeenranta University of Technology, Finland

Tapani JokinenDepartment of Electrical Engineering, Helsinki University of Technology, Finland

Val ria Hrabovcov e aDepartment of Power Electrical Systems, Faculty of Electrical Engineering, University of Zilina, Slovak Republic

Translated by Hanna Niemel aDepartment of Electrical Engineering, Lappeenranta University of Technology, Finland

This edition rst published 2008 C 2008 John Wiley & Sons, Ltd Adapted from the original version in Finnish written by Juha Pyrh nen and published by Lappeenranta University o of Technology C Juha Pyrh nen, 2007 o Registered ofce John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial ofces, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the authors to be identied as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Pyrh nen, Juha. o Design of rotating electrical machines / Juha Pyrh nen, Tapani Jokinen, Val ria Hrabovcov ; translated by o e a Hanna Niemel . a p. cm. Includes bibliographical references and index. ISBN 978-0-470-69516-6 (cloth) 1. Electric machineryDesign and construction. 2. Electric generatorsDesign and construction. 3. Electric motorsDesign and construction. 4. Rotational motion. I. Jokinen, Tapani, 1937 II. Hrabovcov , Val ria. a e III. Title. TK2331.P97 2009 621.31 042dc22 2008042571 A catalogue record for this book is available from the British Library. ISBN: 978-0-470-69516-6 (H/B) Typeset in 10/12pt Times by Aptara Inc., New Delhi, India. Printed in Great Britain by CPI Antony Rowe, Chippenham, Wiltshire

ContentsAbout the Authors Preface Abbreviations and Symbols 1 1.1 1.2 1.3 Principal Laws and Methods in Electrical Machine Design Electromagnetic Principles Numerical Solution The Most Common Principles Applied to Analytic Calculation 1.3.1 Flux Line Diagrams 1.3.2 Flux Diagrams for Current-Carrying Areas Application of the Principle of Virtual Work in the Determination of Force and Torque Maxwells Stress Tensor; Radial and Tangential Stress Self-Inductance and Mutual Inductance Per Unit Values Phasor Diagrams Bibliography xi xiii xv 1 1 9 12 17 22 25 33 36 40 43 45 47 48 48 52 53 54 56 63 71 72 81 92 95 95 98

1.4 1.5 1.6 1.7 1.8

2 Windings of Electrical Machines 2.1 Basic Principles 2.1.1 Salient-Pole Windings 2.1.2 Slot Windings 2.1.3 End Windings 2.2 Phase Windings 2.3 Three-Phase Integral Slot Stator Winding 2.4 Voltage Phasor Diagram and Winding Factor 2.5 Winding Analysis 2.6 Short Pitching 2.7 Current Linkage of a Slot Winding 2.8 Poly-Phase Fractional Slot Windings 2.9 Phase Systems and Zones of Windings 2.9.1 Phase Systems 2.9.2 Zones of Windings

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2.10 Symmetry Conditions 2.11 Base Windings 2.11.1 First-Grade Fractional Slot Base Windings 2.11.2 Second-Grade Fractional Slot Base Windings 2.11.3 Integral Slot Base Windings 2.12 Fractional Slot Windings 2.12.1 Single-Layer Fractional Slot Windings 2.12.2 Double-Layer Fractional Slot Windings 2.13 Single- and Two-Phase Windings 2.14 Windings Permitting a Varying Number of Poles 2.15 Commutator Windings 2.15.1 Lap Winding Principles 2.15.2 Wave Winding Principles 2.15.3 Commutator Winding Examples, Balancing Connectors 2.15.4 AC Commutator Windings 2.15.5 Current Linkage of the Commutator Winding and Armature Reaction 2.16 Compensating Windings and Commutating Poles 2.17 Rotor Windings of Asynchronous Machines 2.18 Damper Windings Bibliography

99 102 103 104 104 105 105 115 122 126 127 131 134 137 140 142 145 147 150 152

3

Design of Magnetic Circuits 3.1 Air Gap and its Magnetic Voltage 3.1.1 Air Gap and Carter Factor 3.1.2 Air Gaps of a Salient-Pole Machine 3.1.3 Air Gap of Nonsalient-Pole Machine 3.2 Equivalent Core Length 3.3 Magnetic Voltage of a Tooth and a Salient Pole 3.3.1 Magnetic Voltage of a Tooth 3.3.2 Magnetic Voltage of a Salient Pole 3.4 Magnetic Voltage of Stator and Rotor Yokes 3.5 No-Load Curve, Equivalent Air Gap and Magnetizing Current of the Machine 3.6 Magnetic Materials of a Rotating Machine 3.6.1 Characteristics of Ferromagnetic Materials 3.6.2 Losses in Iron Circuits 3.7 Permanent Magnets in Rotating Machines 3.7.1 History and Characteristics of Permanent Magnets 3.7.2 Operating Point of a Permanent Magnet Circuit 3.7.3 Application of Permanent Magnets in Electrical Machines 3.8 Assembly of Iron Stacks 3.9 Magnetizing Inductance Bibliography

153 159 159 164 169 171 173 173 177 177 180 183 187 193 200 200 205 213 219 221 224

Contents

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4

Flux Leakage 4.1 Division of Leakage Flux Components 4.1.1 Leakage Fluxes Not Crossing an Air Gap 4.1.2 Leakage Fluxes Crossing an Air Gap 4.2 Calculation of Flux Leakage 4.2.1 Air-Gap Leakage Inductance 4.2.2 Slot Leakage Inductance 4.2.3 Tooth Tip Leakage Inductance 4.2.4 End Winding Leakage Inductance 4.2.5 Skewing Factor and Skew Leakage Inductance Bibliography Resistances 5.1 DC Resistance 5.2 Inuence of Skin Effect on Resistance 5.2.1 Analytical Calculation of Resistance Factor 5.2.2 Critical Conductor Height 5.2.3 Methods to Limit the Skin Effect 5.2.4 Inductance Factor 5.2.5 Calculation of Skin Effect Using Circuit Analysis 5.2.6 Double-Sided Skin Effect Bibliography Main Dimensions of a Rotating Machine 6.1 Mechanical Loadability 6.2 Electrical Loadability 6.3 Magnetic Loadability 6.4 Air Gap Bibliography Design Process and Properties of Rotating Electrical Machines 7.1 Asynchronous Motor 7.1.1 Current Linkage and Torque Production of an Asynchronous Machine 7.1.2 Impedance and Current Linkage of a Cage Winding 7.1.3 Characteristics of an Induction Machine 7.1.4 Equivalent Circuit Taking Asynchronous Torques and Harmonics into Account 7.1.5 Synchronous Torques 7.1.6 Selection of the Slot Number of a Cage Winding 7.1.7 Construction of an Induction Motor 7.1.8 Cooling and Duty Types 7.1.9 Examples of the Parameters of Three-Phase Industrial Induction Motors

225 227 227 228 230 230 234 245 246 250 253 255 255 256 256 265 266 267 267 274 280 281 291 293 294 297 300 301 313 315 320 327 332 337 339 342 343 348

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6

7

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Contents

7.1.10 Asynchronous Generator 7.1.11 Asynchronous Motor Supplied with Single-Phase Current 7.2 Synchronous Machine 7.2.1 Inductances of a Synchronous Machine in Synchronous Operation and in Transients 7.2.2 Loaded Synchronous Machine and Load Angle Equation 7.2.3 RMS Value Phasor Diagrams of a Synchronous Machine 7.2.4 No-Load Curve and Short-Circuit Test 7.2.5 Asynchronous Drive 7.2.6 Asymmetric-Load-Caused Damper Currents 7.2.7 Shift of Damper Bar Slotting from the Symmetry Axis of the Pole 7.2.8 V Curve of a Synchronous Machine 7.2.9 Excitation Methods of a Synchronous Machine 7.2.10 Permanent Magnet Synchronous Machines 7.2.11 Synchronous Reluctance Machines 7.3 DC Machines 7.3.1 Conguration of DC Machines 7.3.2 Operation and Voltage of a DC Machine 7.3.3 Armature Reaction of a DC Machine and Machine Design 7.3.4 Commutation 7.4 Doubly Salient Reluctance Machine 7.4.1 Operating Principle of a Doubly Salient Reluctance Machine 7.4.2 Torque of an SR Machine 7.4.3 Operation of an SR Machine 7.4.4 Basic Terminology, Phase Number and Dimensioning of an SR Machine 7.4.5 Control Systems of an SR Motor 7.4.6 Future Scenarios for SR Machines Bibliography 8 Insulation of Electrical Machines 8.1 Insulation of Rotating Electrical Machines 8.2 Impregnation Varnishes and Resins 8.3 Dimensioning of an Insulation 8.4 Electrical Reactions Ageing Insulation 8.5 Practical Insulation Constructions 8.5.1 Slot Insulations of Low-Voltage Machines 8.5.2 Coil End Insulations of Low-Voltage Machines 8.5.3 Pole Winding Insulations 8.5.4 Low-Voltage Machine Impregnation 8.5.5 Insulation of High-Voltage Machines 8.6 Condition Monitoring of Insulation 8.7 Insulation in Frequency Converter Drives Bibliography

351 353 358 359 370 376 383 386 391 392 394 394 395 400 404 404 405 409 411 413 414 415 416 419 422 425 427 429 431 436 440 443 444 445 445 446 447 447 449 453 455

Contents

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9

Heat Transfer 9.1 Losses 9.1.1 Resistive Losses 9.1.2 Iron Losses 9.1.3 Additional Losses 9.1.4 Mechanical Losses 9.2 Heat Removal 9.2.1 Conduction 9.2.2 Radiation 9.2.3 Convection 9.3 Thermal Equivalent Circuit 9.3.1 Analogy between Electrical and Thermal Quantities 9.3.2 Average Thermal Conductivity of a Winding 9.3.3 Thermal Equivalent Circuit of an Electrical Machine 9.3.4 Modelling of Coolant Flow 9.3.5 Solution of Equivalent Circuit 9.3.6 Cooling Flow Rate Bibliography

457 458 458 460 460 460 462 463 466 470 476 476 477 479 488 493 495 496 497 501 503

Appendix A Appendix B Index

About the AuthorsJuha Pyrh nen is a Professor in the Department of Electrical Engineering at Lappeenranta o University of Technology, Finland. He is engaged in the research and development of electric motors and drives. He is especially active in the elds of permanent magnet synchronous machines and drives and solid-rotor high-speed induction machines and drives. He has worked on many research and industrial development projects and has produced numerous publications and patents in the eld of electrical engineering. Tapani Jokinen is a Professor Emeritus in the Department of Electrical Engineering at Helsinki University of Technology, Finland. His principal research interests are in AC machines, creative problem solving and product development processes. He has worked as an electrical machine design engineer with Oy Str mberg Ab Works. He has been a consulo tant for several companies, a member of the Board of High Speed Tech Ltd and Neorem Magnets Oy, and a member of the Supreme Administrative Court in cases on patents. His research projects include, among others, the development of superconducting and large permanent magnet motors for ship propulsion, the development of high-speed electric motors and active magnetic bearings, and the development of nite element analysis tools for solving electrical machine problems. Val ria Hrabovcov is a Professor of Electrical Machines in the Department of Power e a Electrical Systems, Faculty of Electrical Engineering, at the University of Zilina, Slovak Republic. Her professional and research interests cover all kinds of electrical machines, electronically commutated electrical machines included. She has worked on many research and development projects and has written numerous scientic publications in the eld of electrical engineering. Her work also includes various pedagogical activities, and she has participated in many international educational projects.

PrefaceElectrical machines are almost entirely used in producing electricity, and there are very few electricity-producing processes where rotating machines are not used. In such processes, at least auxiliary motors are usually needed. In distributed energy systems, new machine types play a considerable role: for instance, the era of permanent magnet machines has now commenced. About half of all electricity produced globally is used in electric motors, and the share of accurately controlled motor drives applications is increasing. Electrical drives provide probably the best control properties for a wide variety of processes. The torque of an electric motor may be controlled accurately, and the efciencies of the power electronic and electromechanical conversion processes are high. What is most important is that a controlled electric motor drive may save considerable amounts of energy. In the future, electric drives will probably play an important role also in the traction of cars and working machines. Because of the large energy ows, electric drives have a signicant impact on the environment. If drives are poorly designed or used inefciently, we burden our environment in vain. Environmental threats give electrical engineers a good reason for designing new and efcient electric drives. Finland has a strong tradition in electric motors and drives. Lappeenranta University of Technology and Helsinki University of Technology have found it necessary to maintain and expand the instruction given in electric machines. The objective of this book is to provide students in electrical engineering with an adequate basic knowledge of rotating electric machines, for an understanding of the operating principles of these machines as well as developing elementary skills in machine design. However, due to the limitations of this material, it is not possible to include all the information required in electric machine design in a single book, yet this material may serve as a manual for a machine designer in the early stages of his or her career. The bibliographies at the end of chapters are intended as sources of references and recommended background reading. The Finnish tradition of electrical machine design is emphasized in this textbook by the important co-authorship of Professor Tapani Jokinen, who has spent decades in developing the Finnish machine design profession. An important view of electrical machine design is provided by Professor Val ria Hrabovcov from Slovak Republic, e a which also has a strong industrial tradition. We express our gratitude to the following persons, who have kindly provided material for this book: Dr Jorma Haataja (LUT), Dr Tanja Hedberg (ITT Water and Wastewater AB), Mr Jari J ppinen (ABB), Ms Hanne Jussila (LUT), Dr Panu Kurronen (The Switch Oy), a Dr Janne Nerg (LUT), Dr Markku Niemel (ABB), Dr Asko Parviainen (AXCO Motors Oy), a

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Preface

Mr Marko Rilla (LUT), Dr Pia Salminen (LUT), Mr Ville Sihvo and numerous other colleagues. Dr Hanna Niemel s contribution to this edition and the publication process of the a manuscript is highly acknowledged. Juha Pyrh nen o Tapani Jokinen Val ria Hrabovcov e a

Abbreviations and SymbolsA A A AC AM A1A2 a B Br Bsat B B1B2 b b0c bc bd bdr bds br bs bv b0 C C C1C2 Cf c cp cth CTI cv D DC linear current density [A/m] magnetic vector potential [V s/m] temperature class 105 C alternating current asynchronous machine armature winding of a DC machine number of parallel paths in windings without commutator: per phase, in windings with a commutator: per half armature, diffusivity magnetic ux density, vector [V s/m2 ], [T] remanence ux density [T] saturation ux density [T] temperature class 130 C commutating pole winding of a DC machine width [m] conductor width [m] conductor width [m] tooth width [m] rotor tooth width [m] stator tooth width [m] rotor slot width [m] stator slot width [m] width of ventilation duct [m] slot opening [m] capacitance [F], machine constant, integration constant temperature class >180 C compensating winding of a DC machine friction coefcient specic heat capacity [J/kg K], capacitance per unit of length, factor, divider, constant specic heat capacity of air at constant pressure heat capacity Comparative Tracking Index specic volumetric heat [kJ/K m3 ] electric ux density [C/m2 ], diameter [m] direct current

xvi

Abbreviations and Symbols

Dr Dri Ds Dse D1D2 d dt E Ea E E E E1E2 e e F F F FEA Fg Fm F1F2 f g G G th H Hc , HcB HcJ H h h 0c hc hd hp h p2 hs h yr h ys I IM Ins Io Is

outer diameter of the rotor [m] inner diameter of the rotor [m] inner diameter of the stator [m] outer diameter of the stator [m] series magnetizing winding of a DC machine thickness [m] thickness of the fringe of a pole shoe [m] electromotive force (emf) [V], RMS, electric eld strength [V/m], scalar, elastic modulus, Youngs modulus [Pa] activation energy [J] electric eld strength, vector [V/m] temperature class 120 C irradiation shunt winding of a DC machine electromotive force [V], instantaneous value e(t) Napiers constant force [N], scalar force [N], vector temperature class 155 C nite element analysis geometrical factor magnetomotive force H dl [A], (mmf) separate magnetizing winding of a DC machine or a synchronous machine frequency [Hz], Moody friction factor coefcient, constant, thermal conductance per unit length electrical conductance thermal conductance magnetic eld strength [A/m] coercivity related to ux density [A/m] coercivity related to magnetization [A/m] temperature class 180 C, hydrogen height [m] conductor height [m] conductor height [m] tooth height [m] height of a subconductor [m] height of pole body [m] stator slot height [m] height of rotor yoke [m] height of stator yoke [m] electric current [A], RMS, brush current, second moment of an area, moment of inertia of an area [m4 ] induction motor counter-rotating current (negative-sequence component) [A] current of the upper bar [A] conductor current


xvii

Iu IC IEC Im i J J Jext JM Jsat Js j j K KL k

current of the lower bar, slot current, slot current amount [A] classes of electrical machines International Electrotechnical Commission imaginary part current [A], instantaneous value i(t) moment of inertia [kg m2 ], current density [A/m2 ], magnetic polarization Jacobian matrix moment of inertia of load [kg m2 ] moment of inertia of the motor, [kgm2 ] saturation of polarization [V s/m2 ] surface current, vector [A/m] difference of the numbers of slots per pole and phase in different layers imaginary unit transformation ratio, constant, number of commutator segments inductance ratio connecting factor (coupling factor), correction coefcient, safety factor, ordinal of layers Carter factor kC space factor for copper, space factor for iron kCu , kFe distribution factor kd machine-related constant kE correction factor kFe ,n short-circuit ratio kk skin effect factor for the inductance kL pitch factor kp pitch factor due to coil side shift kpw skin effect factor for the resistance kR saturation factor ksat skewing factor ksq coefcient of heat transfer [W/m2 K] kth pitch factor of the coil side shift in a slot kv winding factor kw safety factor in the yield k L self-inductance [H] L characteristic length, characteristic surface, tube length [m] LC inductorcapacitor tooth tip leakage inductance [H] Ld short-circuit inductance [H] Lk magnetizing inductance [H] Lm magnetizing inductance of an m-phase synchronous machine, in d-axis [H] L md mutual inductance [H] L mn main inductance of a single phase [H] L pd slot inductance [H] Lu transient inductance [H] L subtransient inductance [H] L L1, L2, L3, network phases

xviii


l l l lew lp lpu lw M Msat m m0 N Nf1 Nu Nu1 Nk Np Nv N N Neven Nodd n n nU nv n P Pin PAM PMSM PWM P1 , Pad , P Pr P p pAl p pd Q Q av Qo Q Q

length [m], closed line, distance, inductance per unit of length, relative inductance, gap spacing between the electrodes unit vector collinear to the integration path effective core length [m] average conductor length of winding overhang [m] wetted perimeter of tube [m] inductance as a per unit value length of coil ends [m] mutual inductance [H], magnetization [A/m] saturation magnetization [A/m] number of phases, mass [kg], constant number of turns in a winding, number of turns in series number of coil turns in series in a single pole Nusselt number number of bars of a coil side in the slot number of turns of compensating winding number of turns of one pole pair number of conductors in each side Nondrive end set of integers set of even integers set of odd integers normal unit vector of the surface rotation speed (rotation frequency) [1/s], ordinal of the harmonic (sub), ordinal of the critical rotation speed, integer, exponent number of section of ux tube in sequence number of ventilation ducts number of ux tube power, losses [W] input power [W] pole amplitude modulation permanent magnet synchronous machine (or motor) pulse width modulation additional loss [W] Prandtl number friction loss [W] number of pole pairs, ordinal, losses per core length aluminium content number of pole pairs of a base winding partial discharge electric charge [C], number of slots, reactive power [VA], average number of slots of a coil group number of free slots number of radii in a voltage phasor graph number of slots of a base winding


xix

Q th q qk qm qth R Rbar RM RMS Rm Rth Re Re Recrit RR r r S1S8 S SM SR SyRM Sc Sp Sr S s T Ta Tam Tb tc TEFC TJ Tmec Tu Tv Tl t t tc tr t* U

quantity of heat number of slots per pole and phase, instantaneous charge, q(t) [C] number of slots in a single zone mass ow [kg/s] density of the heat ow [W/m2 ] resistance [ ], gas constant, 8.314 472 [J/K mol], thermal resistance, reactive parts bar resistance [ ] reluctance machine root mean square reluctance [A/V s = 1/H] thermal resistance [K/W] real part Reynolds number critical Reynolds number Resin-rich (impregnation method) radius [m], thermal resistance per unit length radius unit vector duty types apparent power [VA], cross-sectional area synchronous motor switched reluctance synchronous reluctance machine cross-sectional area of conductor [m2 ] pole surface area [m2 ] rotor surface area facing the air gap [m2 ] Poyntings vector [W/m2 ], unit vector of the surface slip, skewing measured as an arc length torque [N m], absolute temperature [K], period [s] Taylor number modied Taylor number pull-out torque, peak torque [N m] commutation period [s] totally enclosed fan-cooled mechanical time constant [s] mechanical torque [N m] pull-up torque [N m] counter torque [N m] locked rotor torque, [N m] time [s], number of phasors of a single radius, largest common divider, lifetime of insulation tangential unit vector commutation period [s] rise time [s] number of layers in a voltage vector graph for a base winding voltage [V], RMS

xx


U Um Usj Uv U1 U2 u u b1 uc um V V Vm VPI V1 V2 v v W W Wd Wfc Wmd Wmt WR W W1 W2 W w X x xm Y Y y ym yn y yv y1 y2 yC

depiction of a phase magnetic voltage [A] peak value of the impulse voltage [V] coil voltage [V] terminal of the head of the U phase of a machine terminal of the end of the U phase of a machine voltage, instantaneous value u(t) [V], number of coil sides in a layer blocking voltage of the oxide layer [V] commutation voltage [V] mean uid velocity in tube [m/s] volume [m3 ], electric potential depiction of a phase scalar magnetic potential [A] vacuum pressure impregnation terminal of the head of the V phase of a machine terminal of the end of the V phase of a machine speed, velocity [m/s] vector energy [J], coil span (width) [m] depiction of a phase energy returned through the diode to the voltage source in SR drives energy stored in the magnetic eld in SR machines energy converted to mechanical work while de-energizing the phase in SR drives energy converted into mechanical work when the transistor is conducting in SR drives energy returning to the voltage source in SR drives coenergy [J] terminal of the head of the W phase of a machine terminal of the end of the W phase of a machine magnetic energy [J] length [m], energy per volume unit reactance [ ] coordinate, length, ordinal number, coil span decrease [m] relative value of reactance admittance [S] temperature class 90 C coordinate, length, step of winding winding step in an AC commutator winding coil span in slot pitches coil span of full-pitch winding in slot pitches (pole pitch expressed in number of slots per pole) coil span decrease in slot pitches step of span in slot pitches, back-end connector pitch step of connection in slot pitches, front-end connector pitch commutator pitch in number of commutator segments


xxi

Z ZM Zs Z0 z za zb zc zp zQ zt 1/ DC i m ph PM r SM str th u z c c D c e ef v T 0 th 0

impedance [ ], number of bars, number of positive and negative phasors of the phase characteristic impedance of the motor [ ] surface impedance [ ] characteristic impedance [ ] coordinate, length, integer, total number of conductors in the armature winding number of adjacent conductors number of brushes number of coils number of parallel-connected conductors number of conductors in a slot number of conductors on top each other angle [rad], [ ], coefcient, temperature coefcient, relative pole width of the pole shoe, convection heat transfer coefcient [W/K] depth of penetration relative pole width coefcient for DC machines factor of the arithmetical average of the ux density mass transfer coefcient [(mol/sm2 )/(mol/m3 ) = m/s] angle between the phase winding relative permanent magnet width heat transfer coefcient of radiation relative pole width coefcient for synchronous machines angle between the phase winding heat transfer coefcient [W/m2 K] slot angle [rad], [ ] phasor angle, zone angle [rad], [ ] angle of single phasor [rad], [ ] angle [rad], [ ], absorptivity energy ratio, integration route interface between iron and air angle [rad], [ ], coefcient commutation angle [rad], [ ] switch conducting angle [rad], [ ] air gap (length), penetration depth [m], dissipation angle [rad], [ ], load angle [rad], [ ] the thickness of concentration boundary layer [m] equivalent air gap (slotting taken into account) [m] effective air gap(inuence of iron taken into account) velocity boundary layer [m] temperature boundary layer [m] load angle [rad], [ ], corrected air gap [m] minimum air gap [m] permittivity [F/m], position angle of the brushes [rad], [ ], stroke angle [rad], [ ], amount of short pitching emissitivity permittivity of vacuum 8.854 1012 [F/m]

xxii


k r 0 A E F Fn Ftan mec SB p q2 r s u v d d0 d0 q q0 th

phase angle [rad], [ ], harmonic factor efciency, empirical constant, experimental pre-exponential constant, reectivity current linkage [A], temperature rise [K] compensating current linkage [A] total current linkage [A] angle [rad], [ ] angle [rad], [ ] angle [rad], [ ], factor for reduction of slot opening, transmissivity permeance, [Vs/A], [H] thermal conductivity [W/m K], permeance factor, proportionality factor, inductance factor, inductance ratio permeability [V s/A m, H/m], number of pole pairs operating simultaneously per phase, dynamic viscosity [Pa s, kg/s m] relative permeability permeability of vacuum, 4 107 [V s/A m, H/m] ordinal of harmonic, Poissons ratio, reluctivity [A m/V s, m/H], pulse velocity reduced conductor height resistivity [ m], electric charge density [C/m2 ], density [kg/m3 ], reection factor, ordinal number of a single phasor absolute overlap ratio effective overlap ratio transformation ratio for IM impedance, resistance, inductance specic conductivity, electric conductivity [S/m], leakage factor, ratio of the leakage ux to the main ux tension [Pa] normal tension [Pa] tangential tension [Pa] mechanical stress [Pa] StefanBoltzmann constant, 5.670 400 108 W/m2 /K4 relative time pole pitch [m] pole pitch on the pole surface [m] rotor slot pitch [m] stator slot pitch [m] slot pitch [m] zone distribution direct-axis transient short-circuit time constant [s] direct-axis transient open-circuit time constant [s] direct-axis subtransient open-circuit time constant [s] quadrature-axis subtransient short-circuit time constant [s] quadrature-axis subtransient open-circuit time constant [s] factor, kinematic viscosity, /, [Pa s/(kg/m3 )] magnetic ux [V s, Wb] thermal power ow, heat ow rate [W] air gap ux [V s], [Wb]


xxiii

T T p Subscripts 0 1 2 Al a ad av B b bar bearing C Cu c contact conv cp cr, crit D DC d EC e ef el em ew ext F Fe f

magnetic ux, instantaneous value (t) [V s], electric potential [V] phase shift angle [rad], [ ] function for skin effect calculation magnetic ux linkage [V s] electric ux [C], function for skin effect calculation length/diameter ratio, shift of a single pole pair mechanical angular speed [rad/s] electric angular velocity [rad/s], angular frequency [rad/s] temperature rise [K, C] temperature gradient [K/m, C/m] pressure drop [Pa]

section primary, fundamental component, beginning of a phase, locked rotor torque, secondary, end of a phase aluminium armature, shaft additional (loss) average brush base value, peak value of torque, blocking bar bearing (losses) capacitor copper conductor, commutation brush contact convection commutating poles critical direct, damper direct current tooth, direct, tooth tip leakage ux eddy current equivalent effective electric electromagnetic end winding external force iron eld

xxiv


Hy i k M m mag max mec min mut N n ns o opt PM p p1 p2 ph ps pu q r res S s sat sj sq str syn tan test th tot u v w x y ya yr ys z

hysteresis internal, insulation compensating, short circuit, ordinal motor mutual, main magnetizing, magnetic maximum mechanical minimum mutual rated nominal, normal negative-sequence component starting, upper optimal permanent magnet pole, primary, subconductor, pole leakage ux pole shoe pole body phasor, phase positive-sequence component per unit quadrature, zone rotor, remanence, relative resultant surface stator saturation impulse wave skew phase section synchronous tangential test thermal total slot, lower, slot leakage ux, pull-up torque zone, coil side shift in a slot, coil end winding leakage ux x-direction y-direction, yoke armature yoke rotor yoke stator yoke z-direction, phasor of voltage phasor graph air gap


xxv

w Subscripts *

ordinal of a subconductor harmonic ordinal number of single phasor friction loss windage (loss) ux leakage ux

A B I I

peak/maximum value, amplitude imaginary, apparent, reduced, virtual base winding, complex conjugate Boldface symbols are used for vectors with components parallel to the unit vectors i, j and k vector potential, A = i Ax + j Ak + k Az ux density, B = i Bx + j Bk + kBz complex phasor of the current bar above the symbol denotes average value

1Principal Laws and Methods in Electrical Machine Design1.1 Electromagnetic PrinciplesA comprehensive command of electromagnetic phenomena relies fundamentally on Maxwells equations. The description of electromagnetic phenomena is relatively easy when compared with various other elds of physical sciences and technology, since all the eld equations can be written as a single group of equations. The basic quantities involved in the phenomena are the following ve vector quantities and one scalar quantity:Electric eld strength Magnetic eld strength Electric ux density Magnetic ux density Current density Electric charge density, dQ/dV E H D B J [V/m] [A/m] [C/m2 ] [V s/m2 ], [T] [A/m2 ] [C/m3 ]

The presence of an electric and magnetic eld can be analysed from the force exerted by the eld on a charged object or a current-carrying conductor. This force can be calculated by the Lorentz force (Figure 1.1), a force experienced by an innitesimal charge dQ moving at a speed v. The force is given by the vector equation dF = dQ(E + v B) = dQE + dQ dl B = dQE + idl B. dt (1.1)

In principle, this vector equation is the basic equation in the computation of the torque for various electrical machines. The latter part of the expression in particular, formulated with a current-carrying element of a conductor of the length dl, is fundamental in the torque production of electrical machines.

Design of Rotating Electrical Machines Juha Pyrh onen, Tapani Jokinen and Valria Hrabovcov e a 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-69516-6

2

Design of Rotating Electrical Machines

i dF

dl

B

Figure 1.1 Lorentz force dF acting on a differential length dl of a conductor carrying an electric current i in the magnetic eld B. The angle is measured between the conductor and the ux density vector B. The vector product i dl B may now be written in the form i dl B = idlB sin

Example 1.1: Calculate the force exerted on a conductor 0.1 m long carrying a current of 10 A at an angle of 80 with respect to a eld density of 1 T. Solution: Using (1.1) we get directly for the magnitude of the force F = |il B| = 10 A 0.1 m sin 80 1 Vs/m2 = 0.98 V A s/m = 0.98 N. In electrical engineering theory, the other laws, which were initially discovered empirically and then later introduced in writing, can be derived from the following fundamental laws presented in complete form by Maxwell. To be independent of the shape or position of the area under observation, these laws are presented as differential equations. A current owing from an observation point reduces the charge of the point. This law of conservation of charge can be given as a divergence equation J= , t (1.2)

which is known as the continuity equation of the electric current. Maxwells actual equations are written in differential form as E = B , t D , t (1.3) (1.4) (1.5) (1.6)

H = J+ D = , B = 0.

Principal Laws and Methods in Electrical Machine Design

3

The curl relation (1.3) of an electric eld is Faradays induction law that describes how a changing magnetic ux creates an electric eld around it. The curl relation (1.4) for magnetic eld strength describes the situation where a changing electric ux and current produce magnetic eld strength around them. This is Amp` res law. Amp` res law also yields a e e law for conservation of charge (1.2) by a divergence Equation (1.4), since the divergence of the curl is identically zero. In some textbooks, the curl operation may also be expressed as E = curl E = rot E. An electric ux always ows from a positive charge and passes to a negative charge. This can be expressed mathematically by the divergence Equation (1.5) of an electric ux. This law is also known as Gausss law for electric elds. Magnetic ux, however, is always a circulating ux with no starting or end point. This characteristic is described by the divergence Equation (1.6) of the magnetic ux density. This is Gausss law for magnetic elds. The divergence operation in some textbooks may also be expressed as D = div D. Maxwells equations often prove useful in their integral form: Faradays induction law E dl =l

d dtS

B dS =

d dt

(1.7)

states that the change of a magnetic ux penetrating an open surface S is equal to a negative line integral of the electric eld strength along the line l around the surface. Mathematically, an element of the surface S is expressed by a differential operator dS perpendicular to the surface. The contour line l of the surface is expressed by a differential vector dl parallel to the line. Faradays law together with Amp` res law are extremely important in electrical machine e design. At its simplest, the equation can be employed to determine the voltages induced in the windings of an electrical machine. The equation is also necessary for instance in the determination of losses caused by eddy currents in a magnetic circuit, and when determining the skin effect in copper. Figure 1.2 illustrates Faradays law. There is a ux penetrating through a surface S, which is surrounded by the line l.

E

B

dS

l

Figure 1.2 Illustration of Faradays induction law. A typical surface S, dened by a closed line l, is penetrated by a magnetic ux with a density B. A change in ux density creates an electric current strength E. The circles illustrate the behaviour of E. dS is a vector perpendicular to the surface S

4


The arrows in the circles point the direction of the electric eld strength E in the case where the ux density B inside the observed area is increasing. If we place a short-circuited metal wire around the ux, we will obtain an integrated voltage l E dl in the wire, and consequently also an electric current. This current creates its own ux that will oppose the ux penetrating through the coil. If there are several turns N of winding (cf. Figure 1.2), the ux does not link all these turns ideally, but with a ratio of less than unity. Hence we may denote the effective turns of winding by kw N, (kw < 1). Equation (1.7) yields a formulation with an electromotive force e of a multi-turn winding. In electrical machines, the factor kw is known as the winding factor (see Chapter 2). This formulation is essential to electrical machines and is written as e = kw N d dtS

B dS = kw N

d d = . dt dt

(1.8)

Here, we introduce the ux linkage = kw N = LI, one of the core concepts of electrical engineering. It may be noted that the inductance L describes the ability of a coil to produce ux linkage . Later, when calculating the inductance, the effective turns, the permeance or the reluctance Rm of the magnetic circuit are needed (L = (kw N)2 = (kw N)2 /Rm ).

Example 1.2: There are 100 turns in a coil having a cross-sectional area of 0.0001 m2 . There is an alternating peak ux density of 1 T linking the turns of the coil with a winding factor of kw = 0.9. Calculate the electromotive force induced in the coil when the ux density variation has a frequency of 100 Hz. Solution: Using Equation (1.8) we get e= d d d = kw N = kw N BS sin t dt dt dt d dt 1 Vs 100 2t 0.0001 m2 sin m2 s

= 0.9 100

e = 90 2 V cos

200 200 t = 565 V cos t. s s

Hence, the peak value of voltage is 565 V and the effective value of the voltage the induced in the coil is 565 V/ 2 = 400 V. Amp` res law involves a displacement current that can be given as the time derivative of e the electric ux . Amp` res law e H dl =l S

J dS +

d dtS

D dS = i (t) +

de dt

(1.9)


5

B, H

l dSe = S0

J

E dS

ie = S0

E dS

Figure 1.3 Application of Amp` res law in the surroundings of a current-carrying conductor. The line e l denes a surface S, the vector dS being perpendicular to it

indicates that a current i(t) penetrating a surface S and including the change of electric ux has to be equal to the line integral of the magnetic ux H along the line l around the surface S. Figure 1.3 depicts an application of Amp` res law. e The term d dtS

D dS =

de dt

in (1.9) is known as Maxwells displacement current, which ultimately links the electromagnetic phenomena together. The displacement current is Maxwells historical contribution to the theory of electromagnetism. The invention of displacement current helped him to explain the propagation of electromagnetic waves in a vacuum in the absence of charged particles or currents. Equation (1.9) is quite often presented in its static or quasi-static form, which yields H dl =l S

J dS =

i (t) = (t) .

(1.10)

The term quasi-static indicates that the frequency f of the phenomenon in question is low enough to neglect Maxwells displacement current. The phenomena occurring in electrical machines meet the quasi-static requirement well, since, in practice, considerable displacement currents appear only at radio frequencies or at low frequencies in capacitors that are deliberately produced to take advantage of the displacement currents. The quasi-static form of Amp` res law is a very important equation in electrical machine e design. It is employed in determining the magnetic voltages of an electrical machine and the required current linkage. The instantaneous value of the current sum i (t) in Equation (1.10), that is the instantaneous value of current linkage , can, if desired, be assumed to involve also the apparent current linkage of a permanent magnet PM = Hc h PM . Thus, the apparent current linkage of a permanent magnet depends on the calculated coercive force Hc of the material and on the thickness hPM of the magnetic material.

6


The corresponding differential form of Amp` res law (1.10) in a quasi-static state (dD/dt e neglected) is written as H = J. The continuity Equation (1.2) for current density in a quasi-static state is written as J = 0. Gausss law for electric elds in integral form D dS =S V

(1.11)

(1.12)

V dV

(1.13)

indicates that a charge inside a closed surface S that surrounds a volume V creates an electric ux density D through the surface. Here V V dV = q (t) is the instantaneous net charge inside the closed surface S. Thus, we can see that in electric elds, there are both sources and drains. When considering the insulation of electrical machines, Equation (1.13) is required. However, in electrical machines, it is not uncommon that charge densities in a medium prove to be zero. In that case, Gausss law for electric elds is rewritten as D dS = 0S

or

D = 0 E = 0.

(1.14)

In uncharged areas, there are no sources or drains in the electric eld either. Gausss law for magnetic elds in integral form B dS = 0S

(1.15)

states correspondingly that the sum of a magnetic ux penetrating a closed surface S is zero; in other words, the ux entering an object must also leave the object. This is an alternative way of expressing that there is no source for a magnetic ux. In electrical machines, this means for instance that the main ux encircles the magnetic circuit of the machine without a starting or end point. Similarly, all other ux loops in the machine are closed. Figure 1.4 illustrates the surfaces S employed in integral forms of Maxwells equations, and Figure 1.5, respectively, presents an application of Gausss law for a closed surface S. The permittivity, permeability and conductivity , and of the medium determine the dependence of the electric and magnetic ux densities and current density on the eld strength. In certain cases, , and can be treated as simple constants; then the corresponding pair of quantities (D and E, B and H, or J and E) are parallel. Media of this kind are called isotropic, which means that , and have the same values in different directions. Otherwise, the media have different values of the quantities , and in different directions, and may therefore be treated as tensors; these media are dened as anisotropic. In practice, the


7

dS S

S dS V

dl (a) (b)

Figure 1.4 Surfaces for the integral forms of the equations for electric and magnetic elds. (a) An open surface S and its contour l, (b) a closed surface S, enclosing a volume V. dS is a differential surface vector that is everywhere normal to the surface

permeability in ferromagnetic materials is always a highly nonlinear function of the eld strength H: = f (H). The general formulations for the equations of a medium can in principle be written as D = f (E), B = f (H), J = f (E). (1.16) (1.17) (1.18)

J E S dS V Q

B

(a)

(b)

Figure 1.5 Illustration of Gausss law for (a) an electric eld and (b) a magnetic eld. The charge Q inside a closed object acts as a source and creates an electric ux with the eld strength E. Correspondingly, a magnetic ux created by the current density J outside a closed surface S passes through the closed surface (penetrates into the sphere and then comes out). The magnetic eld is thereby sourceless (div B = 0)

8


The specic forms for the equations have to be determined empirically for each medium in question. By applying permittivity [F/m], permeability [V s/A m] and conductivity [S/m], we can describe materials by the following equations: D = E, B = H, J = E. (1.19) (1.20) (1.21)

The quantities describing the medium are not always simple constants. For instance, the permeability of ferromagnetic materials is strongly nonlinear. In anisotropic materials, the direction of ux density deviates from the eld strength, and thus and can be tensors. In a vacuum the values are 0 = 8.854 1012 F/m, A s/V m and 0 = 4 107 H/m, V s/A m.

Example 1.3: Calculate the electric eld density D over an insulation layer 0.3 mm thick when the potential of the winding is 400 V and the magnetic circuit of the system is at earth potential. The relative permittivity of the insulation material is r = 3. Solution: The electric eld strength across the insulation is E = 400 V/0.3 mm = 133 kV/m. According to Equation (1.19), the electric eld density is D = E = r 0 E = 3 8.854 1012 A s/V m 133 kV/m = 3.54 A s/m2 .

Example 1.4: Calculate the displacement current over the slot insulation of the previous example at 50 Hz when the insulation surface is 0.01 m2 . Solution: The electric eld over the insulation is e = DS = 0.0354 A s. The time-dependent electric eld over the slot insulation is e (t) = e sin t = 0.0354 A s sin 314t. Differentiating with respect to time gives de (t) = e cos t = 11 A cos 314t. dt The effective current over the insulation is hence 11/ 2 = 7.86 A. Here we see that the displacement current is insignicant from the viewpoint of the machines basic functionality. However, when a motor is supplied by a frequency converter and


9

the transistors create high frequencies, signicant displacement currents may run across the insulation and bearing current problems, for instance, may occur.

1.2 Numerical SolutionThe basic design of an electrical machine, that is the dimensioning of the magnetic and electric circuits, is usually carried out by applying analytical equations. However, accurate performance of the machine is usually evaluated using different numerical methods. With these numerical methods, the effect of a single parameter on the dynamical performance of the machine can be effectively studied. Furthermore, some tests, which are not even feasible in laboratory circumstances, can be virtually performed. The most widely used numerical method is the nite element method (FEM), which can be used in the analysis of two- or three-dimensional electromagnetic eld problems. The solution can be obtained for static, time-harmonic or transient problems. In the latter two cases, the electric circuit describing the power supply of the machine is coupled with the actual eld solution. When applying FEM in the electromagnetic analysis of an electrical machine, special attention has to be paid to the relevance of the electromagnetic material data of the structural parts of the machine as well as to the construction of the nite element mesh. Because most of the magnetic energy is stored in the air gap of the machine and important torque calculation formulations are related to the air-gap eld solution, the mesh has to be sufciently dense in this area. The rule of thumb is that the air-gap mesh should be divided into three layers to achieve accurate results. In the transient analysis, that is in time-stepping solutions, the selection of the size of the time step is also important in order to include the effect of high-order time harmonics in the solution. A general method is to divide one time cycle into 400 steps, but the division could be even denser than this, in particular with highspeed machines. There are ve common methods to calculate the torque from the FEM eld solution. The solutions are (1) the Maxwell stress tensor method, (2) Arkkios method, (3) the method of magnetic coenergy differentiation, (4) Coulombs virtual work and (5) the magnetizing current method. The mathematical torque formulations related to these methods will shortly be discussed in Sections 1.4 and 1.5. The magnetic elds of electrical machines can often be treated as a two-dimensional case, and therefore it is quite simple to employ the magnetic vector potential in the numerical solution of the eld. In many cases, however, the elds of the machine are clearly three dimensional, and therefore a two-dimensional solution is always an approximation. In the following, rst, the full three-dimensional vector equations are applied. The magnetic vector potential A is given by B = A; (1.22)

Coulombs condition, required to dene unambiguously the vector potential, is written as A = 0. (1.23)

10


The substitution of the denition for the magnetic vector potential in the induction law (1.3) yields E = A. t (1.24)

Electric eld strength can be expressed by the vector potential A and the scalar electric potential as E= A t (1.25)

where is the reduced electric scalar potential. Because 0, adding a scalar potential causes no problems with the induction law. The equation shows that the electric eld strength vector consists of two parts, namely a rotational part induced by the time dependence of the magnetic eld, and a nonrotational part created by electric charges and the polarization of dielectric materials. Current density depends on the electric eld strength J = E = A . t (1.26)

Amp` res law and the denition for vector potential yield e Substituting (1.26) into (1.27) gives A 1 A + + = 0. t (1.28) 1 A = J. (1.27)

The latter is valid in areas where eddy currents may be induced, whereas the former is valid in areas with source currents J = Js , such as winding currents, and areas without any current densities J = 0. In electrical machines, a two-dimensional solution is often the obvious one; in these cases, the numerical solution can be based on a single component of the vector potential A. The eld solution (B, H) is found in an xy plane, whereas J, A and E involve only the z-component. The gradient only has a z-component, since J and A are parallel to z, and (1.26) is valid. The reduced scalar potential is thus independent of x- and y-components. could be a linear function of the z-coordinate, since a two-dimensional eld solution is independent of z. The assumption of two-dimensionality is not valid if there are potential differences caused by electric charges or by the polarization of insulators. For two-dimensional cases with eddy currents, the reduced scalar potential has to be set as = 0.


11

In a two-dimensional case, the previous equation is rewritten as Az 1 Az + = 0. t (1.29)

Outside eddy current areas, the following is valid: 1 Az = Jz . (1.30)

The denition of vector potential yields the following components for ux density: Bx = Az , y By = Az . x (1.31)

Hence, the vector potential remains constant in the direction of the ux density vector. Consequently, the iso-potential curves of the vector potential are ux lines. In the two-dimensional case, the following formulation can be obtained from the partial differential equation of the vector potential: k x Az x + y Az y = kJ. (1.32)

Here is the reluctivity of the material. This again is similar to the equation for a static electric eld (A) = J. (1.33)

Further, there are two types of boundary conditions. Dirichlets boundary condition indicates that a known potential, here the known vector potential A = constant, (1.34)

can be achieved for a vector potential for instance on the outer surface of an electrical machine. The eld is parallel to the contour of the surface. Similar to the outer surface of an electrical machine, also the centre line of the machines pole can form a symmetry plane. Neumanns homogeneous boundary condition determined with the vector potential A =0 n (1.35)

can be achieved when the eld meets a contour perpendicularly. Here n is the normal unit vector of a plane. A contour of this kind is for instance part of a eld conned to innite permeability iron or the centre line of the pole clearance.

12


l

Dirichlet

12

A is constant, corresponds to a flux line y

Neumann

A n

=0

A1,A

2

z x A is constant, corresponds to Dirichlets boundary condition

Figure 1.6 Left, a two-dimensional eld and its boundary conditions for a salient-pole synchronous machine are illustrated. Here, the constant value of the vector potential A (e.g. the machines outer contour) is taken as Dirichlets boundary condition, and the zero value of the derivative of the vector potential with respect to normal is taken as Neumanns boundary condition. In the case of magnetic scalar potential, the boundary conditions with respect to potential would take opposite positions. Because of symmetry, the zero value of the normal derivative of the vector potential corresponds to the constant magnetic potential V m , which in this case would be a known potential and thus Dirichlets boundary condition. Right, a vector-potential-based eld solution of a two-pole asynchronous machine assuming a two-dimensional eld is presented

The magnetic ux penetrating a surface is easy to calculate with the vector potential. Stokes theorem yields for the ux =S

B dS =S

( A) dS =l

A dl.

(1.36)

This is an integral around the contour l of the surface S. These phenomena are illustrated with Figure 1.6. In the two-dimensional case of the illustration, the end faces share of the integral is zero, and the vector potential along the axis is constant. Consequently, for a machine of length l we obtain a ux 12 = l (A1 A2 ) . This means that the ux 12 is the ux between vector equipotential lines A1 and A2 . (1.37)

1.3 The Most Common Principles Applied to Analytic CalculationThe design of an electrical machine involves the quantitative determination of the magnetic ux of the machine. Usually, phenomena in a single pole are analysed. In the design of a magnetic circuit, the precise dimensions for individual parts are determined, the required current linkage for the magnetic circuit and also the required magnetizing current are calculated, and the magnitude of losses occurring in the magnetic circuit are estimated.


13

If the machine is excited with permanent magnets, the permanent magnet materials have to be selected and the main dimensions of the parts manufactured from these materials have to be determined. Generally, when calculating the magnetizing current for a rotating machine, the machine is assumed to run at no load: that is, there is a constant current owing in the magnetizing winding. The effects of load currents are analysed later. The design of a magnetic circuit of an electrical machine is based on Amp` res law (1.4) e and (1.8). The line integral calculated around the magnetic circuit of an electrical machine, that is the sum of magnetic potential differences Um,i , is equal to the surface integral of the current densities over the surface S of the magnetic circuit. (The surface S here indicates the surface penetrated by the main ux.) In practice, in electrical machines, the current usually ows in the windings, the surface integral of the current density corresponding to the sum of these currents (owing in the windings), that is the current linkage . Now Amp` res law can e be rewritten as Um,tot = Um,i =l

H dl =S

J dS = =

i.

(1.38)

The sum of magnetic potential differences U m around the complete magnetic circuit is equal to the sum of the magnetizing currents in the circuit, that is the current linkage . In simple applications, the current sum may be given as i = kw N i, where kw N is the effective number of turns and i the current owing in them. In addition to the windings, this current linkage may also involve the effect of the permanent magnets. In practice, when calculating the magnetic voltage, the machine is divided into its components, and the magnetic voltage U m between points a and b is determined as Um,ab =a b

H dl.

(1.39)

In electrical machines, the eld strength is often in the direction of the component in question, and thus Equation (1.39) can simply be rewritten as Um,ab =a b

H dl.

(1.40)

Further, if the eld strength is constant in the area under observation, we get Um,ab = Hl. (1.41)

In the determination of the required current linkage of a machines magnetizing winding, the simplest possible integration path is selected in the calculation of the magnetic voltages. This means selecting a path that encloses the magnetizing winding. This path is dened as the main integration path and it is also called the main ux path of the machine (see Chapter 3). In salient-pole machines, the main integration path crosses the air gap in the middle of the pole shoes.

14


Example 1.5: Consider a C-core inductor with a 1 mm air gap. In the air gap, the ux density is 1 T. The ferromagnetic circuit length is 0.2 m and the relative permeability of the core material at 1 T is r = 3500. Calculate the eld strengths in the air gap and the core, and also the magnetizing current needed. How many turns N of wire carrying a 10 A direct current are needed to magnetize the choke to 1 T? Fringing in the air gap is neglected and the winding factor is assumed to be kw = 1. Solution: According to (1.20), the magnetic eld strength in the air gap is H = B /0 = 1 V s/m2 / 4 107 V s/A m = 795 kA/m. The corresponding magnetic eld strength in the core is HFe = BFe (r 0 ) = 1 V s/m2 3500 4 107 V s/A m = 227 A/m.

The magnetic voltage in the air gap (neglecting fringing) is Um, = H = 795 kA/m 0.001 m = 795 A. The magnetic voltage in the core is Um,Fe = HFelFe = 227 A/m 0.2 m = 45 A. The magnetomotive force (mmf) of the magnetic circuit is H dl = Um,tot =l

Um,i = Um, + Um,Fe = 795 A + 45 A = 840 A.

The current linkage of the choke has to be of equal magnitude with the mmf U m,tot , = We get N= Um,tot 840 A = = 84 turns. kw i 1 10 A i = kw N i = Um,tot .

In machine design, not only does the main ux have to be analysed, but also all the leakage uxes of the machine have to be taken into account. In the determination of the no-load curve of an electrical machine, the magnetic voltages of the magnetic circuit have to be calculated with several different ux densities. In practice, for the exact denition of the magnetizing curve, a computation program that solves the different magnetizing states of the machine is required. According to their magnetic circuits, electrical machines can be divided into two main categories: in salient-pole machines, the eld windings are concentrated pole windings, whereas in


15

nonsalient-pole machines, the magnetizing windings are spatially distributed in the machine. The main integration path of a salient-pole machine consists for instance of the following components: a rotor yoke (yr), pole body (p2), pole shoe (p1), air gap (), teeth (d) and armature yoke (ya). For this kind of salient-pole machine or DC machine, the total magnetic voltage of the main integration path therefore consists of the following components Um,tot = Um,yr + 2Um,p2 + 2Um,p1 + 2Um, + 2Um,d + Um,ya . (1.42)

In a nonsalient-pole synchronous machine and induction motor, the magnetizing winding is contained in slots. Therefore both stator (s) and rotor (r) have teeth areas (d) Um,tot = Um,yr + 2Um,dr + 2Um, + 2Um,ds + Um,ys . (1.43)

With Equations (1.42) and (1.43), we must bear in mind that the main ux has to ow twice across the teeth area (or pole arc and pole shoe) and air gap. In a switched reluctance (SR) machine, where both the stator and rotor have salient poles (double saliency), the following equation is valid: Um,tot = Um,yr + 2Um,rp2 + 2Um,rp1 () + 2Um, () + 2Um,sp1 () + 2Um,sp2 + Um,ys . (1.44) This equation proves difcult to employ, because the shape of the air gap in an SR machine varies constantly when the machine rotates. Therefore the magnetic voltage of both the rotor and stator pole shoes depends on the position of the rotor . The magnetic potential differences of the most common rotating electrical machines can be presented by equations similar to Equations (1.42)(1.44). In electrical machines constructed of ferromagnetic materials, only the air gap can be considered magnetically linear. All ferromagnetic materials are both nonlinear and often anisotropic. In particular, the permeability of oriented electrical steel sheets varies in different directions, being highest in the rolling direction and lowest in the perpendicular direction. This leads to a situation where the permeability of the material is, strictly speaking, a tensor. The ux is a surface integral of the ux density. Commonly, in electrical machine design, the ux density is assumed to be perpendicular to the surface to be analysed. Since the area of a perpendicular surface S is S, we can rewrite the equation simply as = BdS. (1.45)

Further, if the ux density B is also constant, we obtain = BS. (1.46)

Using the equations above, it is possible to construct a magnetizing curve for each part of the machine ab = f Um,ab , B = f Um,ab . (1.47)

16


In the air gap, the permeability is constant = 0 . Thus, we can employ magnetic conductivity, that is permeance , which leads us to ab = ab Um,ab . If the air gap eld is homogeneous, we get ab = ab Um,ab = 0 S Um,ab . (1.49) (1.48)

Equations (1.38) and (1.42)(1.44) yield the magnetizing curve for a machine = f (), B = f (), (1.50)

where the term is the air-gap ux. The absolute value for ux density B is the maximum ux density in the air gap in the middle of the pole shoe, when slotting is neglected. The magnetizing curve of the machine is determined in the order , B B H Um by always selecting a different value for the air-gap ux , or for its density, and by calculating the magnetic voltages in the machine and the required current linkage . With the current linkage, it is possible to determine the current I owing in the windings. Correspondingly, with the air-gap ux and the winding, we can determine the electromotive force (emf) E induced in the windings. Now we can nally plot the actual no-load curve of the machine (Figure 1.7) E = f (I ). (1.51)

E

0 0 Im

0 0 Im

Figure 1.7 Typical no-load curve for an electrical machine expressed by the electromotive force E or the ux linkage as a function of the magnetizing current I m . The E curve as a function of I m has been measured when the machine is running at no load at a constant speed. In principle, the curve resembles a BH curve of the ferromagnetic material used in the machine. The slope of the no-load curve depends on the BH curve of the material, the (geometrical) dimensions and particularly on the length of the air gap


17

3

S3

S2

2

S1

1

Figure 1.8 Laminated tooth and a coarse ux tube running in a lamination. The cross-sections of the tube are presented with surface vectors Si . There is a ux owing in the tube. The ux tubes follow the ux lines in the magnetic circuit of the electrical machine. Most of the tubes constitute the main magnetic circuit, but a part of the ux tubes forms leakage ux paths. If a two-dimensional eld solution is assumed, two-dimensional ux diagrams as shown in Figure 1.6 may replace the ux tube approach

1.3.1 Flux Line DiagramsLet us consider areas with an absence of currents. A spatial magnetic ux can be assumed to ow in a ux tube. A ux tube can be analysed as a tube of a quadratic cross-section S. The ux does not ow through the walls of the tube, and hence B dS = 0 is valid for the walls. As depicted in Figure 1.8, we can see that the corners of the ux tube form the ux lines. When calculating a surface integral along a closed surface surrounding the surface of a ux tube, Gausss law (1.15) yields B dS = 0. (1.52)

Since there is no ux through the side walls of the tube in Figure 1.8, Equation (1.52) can be rewritten as B1 d S1 = B2 d S2 = B3 d S3 , (1.53)

18


indicating that the ux of the ux tube is constant 1 = 2 = 3 = . (1.54)

A magnetic equipotential surface is a surface with a certain magnetic scalar potential V m . When travelling along any route between two points a and b on this surface, we must getb

H dl = Um,ab = Vma Vmb = 0.a

(1.55)

When observing a differential route, this is valid only when H dl = 0. For isotropic materials, the same result can be expressed as B dl = 0. In other words, the equipotential surfaces are perpendicular to the lines of ux. If we select an adequately small area S of the surface S, we are able to calculate the ux = B S. (1.56)

The magnetic potential difference between two equipotential surfaces that are close enough to each other (H is constant along the integration path l) is written as Um = Hl. The above equations give the permeance of the cross-section of the ux tube = dS B dS = . = Um Hl l (1.58) (1.57)

The ux line diagram (Figure 1.9) comprises selected ux and potential lines. The selected ux lines conne ux tubes, which all have an equal ux . The magnetic voltage between the chosen potential lines is always the same, U m . Thus, the magnetic conductivity of each section of the ux tube is always the same, and the ratio of the distance of ux lines x to the distance of potential lines y is always the same. If we set x = 1, y (1.59)

the eld diagram forms, according to Figure 1.9, a grid of quadratic elements. In a homogeneous eld, the eld strength H is constant at every point of the eld. According to Equations (1.57) and (1.59), the distance of all potential and ux lines is thus always the same. In that case, the ux diagram comprises squares of equal size. When constructing a two-dimensional orthogonal eld diagram, for instance for the air gap of an electrical machine, certain boundary conditions have to be known to be able to draw the diagram. These boundary conditions are often solved based on symmetry, or also because the potential of a certain potential surface of the ux tube in Figure 1.8 is already known. For instance, if the stator and rotor length of the machine is l, the area of the ux tube can,


19

S1 z

flux

s line

x y

Um

Vm

2 1

Vm potential line

S2

Figure 1.9 Flux lines and potential lines in a three-dimensional area with a ux owing across an area where the length dimension z is constant. In principle, the diagram is thus two dimensional. Such a diagram is called an orthogonal eld diagram

without signicant error, be written as dS = l dx. The interface of the iron and air is now analysed according to Figure 1.10a. We get d y = B yldx B yFeldx = 0 B y = B yFe . (1.60)

Here, By and ByFe are the ux densities of air in the y-direction and of iron in the y-direction.

y y By Bx air air gap 0 By Bx iron, e.g. rotor surface z Fe (a) dxy

dy 0 l x z iron Fe (b) x

Figure 1.10 (a) Interface of air and iron Fe. The x-axis is tangential to the rotor surface. (b) Flux travelling on iron surface

20


In Figure 1.10a, the eld strength has to be continuous in the x-direction on the ironair interface. If we consider the interface in the x-direction and, based on Amp` res law, assume e a section dx of the surface has no current, we get Hx dx HxFe dx = 0, and thus BxFe . Fe (1.61)

Hx = HxFe =

(1.62)

By assuming that the permeability of iron is innite, Fe , we get HxFe = Hx = 0 and thereby also Bx = 0 Hx = 0. Hence, if we set Fe , the ux lines leave the ferromagnetic material perpendicularly into the air. Simultaneously, the interface of iron and air forms an equipotential surface. If the iron is not saturated, its permeability is very high, and the ux lines can be assumed to leave the iron almost perpendicularly in currentless areas. In saturating areas, the interface of the iron and air cannot strictly be considered an equipotential surface. The magnetic ux and the electric ux refract on the interface. In Figure 1.10b, the ux ows in the iron in the direction of the interface. If the iron is not saturated (Fe ) we can set Bx 0. Now, there is no ux passing from the iron into air. When the iron is about to become saturated (Fe 1), a signicant magnetic voltage occurs in the iron. Now, the air adjacent to the iron becomes an appealing route for the ux, and part of the ux passes into the air. This is the case for instance when the teeth of electrical machines saturate: a part of the ux ows across the slots of the machine, even though the permeability of the materials in the slot is in practice equal to the permeability of a vacuum. The lines of symmetry in ux diagrams are either potential or eld lines. When drawing a ux diagram, we have to know if the lines of symmetry are ux or potential lines. Figure 1.11 is an example of an orthogonal eld diagram, in which the line of symmetry forms a potential line; this could depict for instance the air gap between the contour of an magnetizing pole of a DC machine and the rotor. The solution of an orthogonal eld diagram by drawing is best started at those parts of the geometry where the eld is as homogeneous as possible. For instance, in the case of Figure 1.11, we start from the point where the air gap is at its narrowest. Now, the surface of the magnetizing pole and the rotor surface that is assumed to be smooth form the potential lines together with the surface between the poles. First, the potential lines are plotted at equal distances and, next, the ux lines are drawn perpendicularly to potential lines so that the area under observation forms a grid of quadratic elements. The length of the machine being l, each ux tube created this way carries a ux of . With the eld diagram, it is possible to solve various magnetic parameters of the area under observation. If n is the number (not necessarily an integer) of contiguous ux tubes carrying a ux , and U m is the magnetic voltage between the sections of a ux tube (nU sections in sequence), the permeance of the entire air gap assuming that b = can be


21

symmetry line between poles

symmetry line in the middle of poleH0, B0

DC machine stator pole potential line

Umb

0 U m 023 20 15 10 5

potential line

DC machine rotor surfacex

b0

1

p/2

0

pole pitch

Figure 1.11 Drawing an orthogonal eld diagram in an air gap of a DC machine in the edge zone of a pole shoe. Here, a differential equation for the magnetic scalar potential is solved by drawing. Dirichlets boundary conditions for magnetic scalar potentials created on the surfaces of the pole shoe and the rotor and on the symmetry plane between the pole shoes. The centre line of the pole shoe is set at the origin of the coordinate system. At the origin, the element is dimensioned as 0 , b0 . The and b in different parts of the diagram have different sizes, but the remains the same in all ux tubes. The pole pitch is p . There are about 23.5 ux tubes from the pole surface to the rotor surface in the gure

written as Um n bl0 n = n l. = = = 0 Um n U Um n U Um nU The magnetic eld strength in the enlarged element of Figure 1.11 is H= and correspondingly the magnetic ux density B = 0 Um = . bl (1.65) Um , (1.64)

(1.63)

With Equation (1.56), it is also possible to determine point by point the distribution of ux density on a potential line; in other words, on the surface of the armature or the magnetizing pole. With the notation in Figure 1.11, we get 0 = B0 b0l = (x) = B(x) b(x)l. (1.66)

In the middle of the pole, where the air-gap ux is homogeneous, the ux density is B0 = 0 H0 = 0 Um Um, = 0 . 0 0 (1.67)

22


Thus, the magnitude of ux density as a function of the x-coordinate is B(x) = b0 B0 = b(x) Um, b0 0 . b(x) 0 (1.68)

Example 1.6: What is the permeance of the main ux in Figure 1.11 when the air gap = 0.01 m and the stator stack length is l = 0.1 m? How much ux is created with f = 1000 A? Solution: In the centre of the pole, the orthogonal ux diagram is uniform and we see that 0 and b0 have the same size; 0 = b0 = 2 mm. The permeance of the ux tube in the centre of the pole is 0 = 0 b0l 0.02 m 0.1 m Vs = 0 = 4 108 . 0 0.02 m A

As we can see in Figure 1.11, about 23.5 ux tubes travel from half of the stator pole to the rotor surface. Each of these ux tubes transmits the same amount of ux, and hence the permeance of the whole pole seen by the main ux is = 2 23.5 0 = 47 0 = 47 4 108 V s Vs = 5.9 . A A

If we have f = 1000 A current linkage magnetizing the air gap, we get the ux = f = 5.9 V s 1000 A = 5.9 mV s. A

1.3.2 Flux Diagrams for Current-Carrying AreasLet us rst consider a situation in which an equivalent linear current density A [A/m] covers the area under observation. In principle, the linear current density corresponds to the surface current Js = n H0 induced in a conducting medium by an alternating eld strength H0 outside the surface. The surface normal unit vector is denoted by n. In an electrical machine with windings, the articial surface current, that is the local value of the linear current density A, may for instance be calculated as a current sum owing in a slot divided by the slot pitch. Equivalent linear current density can be employed in approximation, because the currents owing in the windings of electrical machines are usually situated close to the air gap, and the current linkages created by the currents excite mainly the air gaps. Thus, we can set = 0 in the observed area of equivalent linear current density. The utilization of equivalent linear current density simplies the manual calculation of the machine by idealizing the potential surfaces, and does not have a crucial impact on the eld diagram in the areas outside the area of linear current density. Figure 1.12 illustrates an equivalent linear current density. The value for equivalent linear current density A is expressed per unit length in the direction of observation. The linear current density A corresponds to the tangential magnetic eld


23

y Hy dy d 0 linear current density A H yFe symmetry line, Neumann's condition linear current density symmetry line, Neumann's condition

Fe x

pole surface Dirichlet's condition

current flowing (a)

rotor surface, Dirichlet's boundary condition (b)

Figure 1.12 (a) General representation of linear current density A [A/m] and (b) its application to the eld diagram of a magnetizing pole of a DC machine. It is important to note that in the area of the pole body, the potential lines now pass from air to iron. Dirichlets boundary conditions indicate here a known equiscalar potential surface

strength H y . Assuming the permeability of iron to be innite, Amp` res law yields for the e element dy of Figure 1.12a H dl = d = Hyair dy HyFe dy = Ady. Further, this gives us Hyair = A and B yair = 0 A. (1.70) (1.69)

Equation (1.70) indicates that in the case of Figure 1.12 we have a tangential ux density on the pole body surface. The tangential ux density makes the ux lines travel inclined to the pole body surface and not perpendicular to it as in currentless areas. If we assume that the phenomenon is observed on the stator inner surface or on the rotor outer surface, the x-components may be regarded as tangential components and the y-components as normal components. In the air gap , there is a tangential eld strength Hx along the x-component, and a corresponding component of ux density Bx created by the linear current density A. This is essential when considering the force density, the tangential stress Ftan , that generates torque (Maxwell stresses will be discussed later). On iron surfaces with linear current density, the ux lines no longer pass perpendicularly from the iron to the air gap, as Figure 1.12 depicting the eld diagram of a DC machines magnetizing pole also illustrates. The inuence of a magnetizing winding on the pole body is illustrated with the linear current density. Since the magnetizing winding is evenly distributed over the length of the pole body (the linear current density being constant), it can be seen that the potential changes linearly in the area of linear current density in the direction of the height of the pole.

24


Vm3 rotor Vm4 Fe

leakage fluH=0 Ha S H=0o

Vm2

xc S b a Vm2

Hb Ha d S bh Hb

= (Hb Ha)b = Um SJ = Vm3 Vm2g acr e flowin oss the

H dl Hbb+0Hab+0

air gap

Vm2 stator

main

flux lin

Vm0 Vm1

Fe

Figure 1.13 Current-carrying conductor in a slot and its eld diagram. The illustration on the left demonstrates the closed line integral around the surface S; also some ux lines in the iron are plotted. Note that the ux lines travelling across the slot depict leakage ux

As evidence of this we can see that in Figure 1.12 the potential lines starting in the air gap enter the area of linear current density at even distances. In areas with current densities J, the potential lines become gradient lines. This can be seen in Figure 1.13 at points a, b and c. We could assume that the gure illustrates for instance a nonsalient-pole synchronous machine eld winding bar carrying a DC density. The magnetic potential difference between V m4 and V m0 equals the slot current. The gradient lines meet the slot leakage ux lines orthogonally, which means that H dl = 0 along a gradient line. In the gure, we calculate a closed line integral around the area S of the surface S H dl = Vm3 Vm2 =S

J dS,

(1.71)

where we can see that when the current density J and the difference of magnetic potentials U m are constant, the area S of the surface S also has to be constant. In other words, the selected gradient lines conne areas of equal size from the surface S with a constant current density. The gradient lines meet at a single point d, which is called an indifference point. If the current-carrying area is conned by an area with innite permeability, the border line is a potential line and the indifference point is located on this border line. If the permeability of iron is not innite, then d is located in the current-carrying area, as in Figure 1.13. If inside a current-carrying area the line integral is dened for instance around the area S, we can see that the closer to the point d we get, the smaller become the distances between the gradient


25

lines. In order to maintain the same current sum in the observed areas, the heights of the areas S have to be changed. Outside the current-carrying area, the following holds: = Um = 0l h b J dS.S

(1.72)

Inside the area under observation, when a closed line integral according to Equation (1.71) is written only for the area S ( 2 cm and for copper bars Equation (4.56) gives > 2, in which case sinh 2 sin 2 , cosh 2 cos 2 and sinh2 cosh2 , and hence kL 3 . 2 (4.61)

Example 4.4: Repeat Example 4.2 for an aluminium squirrel cage bar at cold start in a 50 Hz supply. Solution: The slot shape is, according to Figure 4.8, b1 = 0.003 m, h1 = 0.002 m, b4 = 0.008 m and h4 = 0.02 m, l = 0.25 m and the slot at height h4 is fully lled with aluminium bars. The conductivity of aluminium at 20 C is Al = 37 MS/m. The permeance factor of the wound part of the slot without skin effect is l4 = h4 0.02 = 0.833. = 3b4 3 0.008

The permeance factor of the slot opening is l1 = h1 0.002 = = 0.667. b1 0.003

The reduced height of the conductor is 0.008 bc = 1.71 = 0.02 2 50 4 107 37 106 2b4 2 0.008

= h 4 0

which is a dimensionless number kL = 1 z2 1 11 ( ) = ( ) ( ) + t 2 ( ) = ( ) + 2 1 zt zt 3 2 sinh 2 sin 2 cosh 2 cos 2 = 3 3.42 sinh 3.42 sin 3.42 cosh 3.42 cos 3.42 = 0.838.

=

Flux Leakage

245

The permeance factor of the slot under skin effect is lu,ec = l1 + kL l4 = 0.667 + 0.838 0.833 = 1.37. This is somewhat less than in Example 4.2. The slot leakage inductance of a squirrel cage bar is2 L u,bar = 0l z Q lu = 4 107 0.25 12 1.37 H = 0.43 H.

4.2.3 Tooth Tip Leakage InductanceThe tooth tip leakage inductance is determined by the magnitude of leakage ux owing in the air gap outside the slot opening. This ux leakage is illustrated in Figure 4.14. The current linkage in the slot causes a potential difference between the teeth on opposite sides of the slot opening, and as a result part of the current linkage will be used to produce the leakage ux of the tooth tip. The tooth tip leakage inductance can be determined by applying a permeance factor 5 ld = k 2 5+4 b1 b1

,

(4.62)

where k2 = (1 + g)/2 is calculated from Equation (4.48). The tooth tip leakage inductance of the whole phase winding is obtained by substituting l d in Equation (4.30): Ld = 4m 0l ld N 2 . Q (4.63)

b1

Figure 4.14 Flux leakage creating a tooth tip leakage inductance around a slot opening

246


In salient-pole machines, we have to substitute the air gap in Equation (4.62) by the air gap at the middle of the pole, where the air gap is at its smallest. If we select the air gap to be innite, we obtain a limit value of l d = 1.25, which is the highest value for l d . If is small, as is the case in asynchronous machines in particular, the inuence of tooth leakage inductance is insignicant. Equations (4.62)(4.63) are no longer valid for the main poles of DC machines. The calculations in the case of DC machines are analysed for instance by Richter (1967).

Example 4.5: Calculate the toot tip leakage of the machine in Example 4.3. The machine is now equipped with rotor surface permanent magnets. The magnets are neodymium iron boron magnets 8 mm thick, and there is a 2 mm physical air gap, p = 2, Q = 24, W/ p = 5/6 and N = 40 (cf. Figure 2.17b). The pole pairs are connected in parallel, a = 2. Compare the result with the slot leakage of Example 4.3. Solution: As the permanent magnets represent, in practice, air (rPM = 1.05), we may assume that the air gap in the calculation of the tooth tip leakage is 2 + 8/1.05 mm = 9.62 mm. The factor k2 = 1 3/4 = 1 3/(4 6) = 0.875. We now obtain for the permeance coefcient 5 ld = k 2 5+4 b1 b1 0.00 962 0.003 = 0.875 0.00 962 5+4 0.003 5

= 0.787.

The tooth tip inductance is, according to Equation (4.63), Ld = 4m 43 0 l ld N 2 = 4 107 0.25 0.787 402 H = 0.198 mH. Q 24

In Example 4.3, the slot leakage of the same machine was 0.340 mH. As the air gap in a rotor surface magnet machine is long, the tooth tip leakage has a signicant value, about 70% of the slot leakage.

4.2.4 End Winding Leakage InductanceEnd winding leakage ux results from all the currents owing in the end windings. The geometry of the end windings is usually difcult to analyse, and, further, all the phases of poly-phase machines inuence the occurrence of a leakage ux. Therefore, the accurate determination of an end winding leakage inductance is a challenging task, which would require a three-dimensional numerical solution. Since the end windings are relatively far from the iron parts, the end winding inductances are not very high, and therefore it sufces to employ empirically determined permeance factors l lew and l w . In machines with an alternating current owing in both the stator and the rotor, measuring always yields a sum of the leakage

Flux Leakage

247

q=2

zQ

zQ

leW

WeWFigure 4.15 Leakage ux and dimensions of an end winding

inductance of the primary winding and the leakage inductance of the secondary winding referred to the primary winding. In a stationary state, there is a direct current owing in the rotor of a synchronous machine, and thus the end winding inductance of a synchronous machine is determined only by the stator side. When calculating the slot leakage inductance, the proportion of each slot can be calculated separately. The number of conductors in a slot is zQ (in a two-layer winding zQ = 2zcs ) and the conductors are surrounded by iron. The ux in the end windings is a result of the inuence of all the coil turns belonging to a coil group. According to Figure 4.15, the number of coil turns is qzQ , which has to replace zQ in the inductance equation (4.23). The average length lw of the end winding (see Figure 4.15) has to be substituted for the length l of the stack in Equation (4.23). The number of coil groups in series in a phase winding is Q/amq, and the number of parallel current paths is a. The equation for end winding leakage inductance can now be written as Lw = Q1 qz Q amq a2

0lw lw =

zQ Q q m a

2

0lw lw =

4m q N 2 0lw lw . Q

(4.64)

The average length lw of the end winding and the product lw l w can be written in the form lw = 2lew + Wew , lw lw = 2lew llew + Wew lW , (4.65) (4.66)

where lew is the axial length of the end winding measured from the end of the stack, and W ew is the coil span according to Figure 4.15. l lew and l w are the corresponding permeance factors, see Tables 4.1 and 4.2. A permeance factor depends on the structure of the winding (e.g. a single-phase, or a twoplane, three-phase, or a three-plane, three-phase, or a diamond winding constructed of coils of equal shapes), the organization of the planes of the end winding, the ratio of the length of

248


Table 4.1 End winding leakage permeance factors of an asynchronous machine for various stator and rotor combinations Type of stator winding Three-phase, three-plane Three-phase, three-plane Three-phase, three-plane Three-phase, two-plane Three-phase two-plane Three-phase, two-plane Cylindrical three-phase diamond winding Cylindrical three-phase diamond winding Single-phase Type of rotor winding Three-phase, three-plane Cylindrical three-phase diamond winding Cage winding Three-phase, two-plane Cylindrical three-phase diamond winding Cage winding Cylindrical three-phase diamond winding Cage winding Cage winding l lew 0.40 0.34 0.34 0.55 0.55 0.50 0.26 0.50 0.23 lw 0.30 0.34 0.24 0.35 0.25 0.20 0.36 0.20 0.13

an average coil end to the pole pitch lw / p , and the rotor type (a nonsalient-pole machine, a salient-pole machine, an armature winding of a DC machine, a cage winding, a three-phase winding). Richter (1967: 279; 1963: 91) and (1954:161) presents in detail some calculated values for permeance factors that are valid for different machine types. Based on the literature, the following tables can be compiled for the denition of the end winding leakage permeance factors for asynchronous and synchronous machines. With these permeance factors, Equation (4.64) gives the sum of the stator end winding leakage inductance and the rotor to the stator referred end winding leakage inductance. The major part of the sum belongs to the stator (6080%).

Table 4.2 Permeance factors of the end windings of a synchronous machine Cross-section of end winding Nonsalient-pole machine l lew 0.342 0.380 0.371 0.493 0.571 0.605 lw 0.413 0.130 0.166 0.074 0.073 0.028 Salient-pole machine l lew 0.297 0.324 0.324 0.440 0.477 0.518 lw 0.232 0.215 0.243 0.170 0.187 0.138

Flux Leakage

249

Example 4.6: The air-gap diameter of the machine in Example 4.3 is 130 mm and the total height of the slots is 22 mm. Calculate the end winding leakage inductance for a three-phase surface-mounted, permanent magnet synchronous machine with Q = 24, q = 2

design of rotating electrical machines - malestrom

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